Re: [math-fun] is your Hurwitz # unlisted?
BG> By uncontrived I meant not explicitly constructed as a continued fraction. NeilB>Just my two cents:http://ttk.pte.hu/mii/html/pannonica/index_elemei/mp17-1/mp17-1-091-110.pdf claims that (1;2,4,8,16...) is equal to 1+Sum[1/((-1)^n*2^(1 + n)^2*QPochhammer[4, 4, n]), {n, 0, Infinity}]/ Sum[1/((-1)^n*2^n^2*QPochhammer[4, 4, n]), {n, 0, Infinity}] (The source, Takao Komatsu's "Hurwitz and Tasoev Continued Fractions with Long Period" cites another paper, "On Hurwitzian and Tasoev's Continued Fractions", by the same author, but which I can't seem to find) --Neil Bickford On Thu, Sep 27, 2012 at 10:33 AM, Allan Wechsler <acwacw@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=acwacw%40gmail.com>> wrote:>> If x is Hurwitz then so is x/2, so, uh ...>> I think that by "contrived" Gosper means "with cf terms explicitly chosen> to be non-periodic". An example would be (1;2,4,8,16...). I'm assuming> that there is not a single such example that anybody knows a "closed form"> for; the closed form thereof would answer Gosper's query.>> On Thu, Sep 27, 2012 at 12:54 PM, meekerdb <meekerdb@verizon.net <http://gosper.org/webmail/src/compose.php?send_to=meekerdb%40verizon.net>> wrote:>> > Is the smallest contrived Hurwitz, thereby uncontrived? :-)> >> > Brent> >> >> > On 9/27/2012 5:38 AM, Bill Gosper wrote:> >>
Can someone name a single (uncontrived) constant (e.g., π, e^3, 2^(1/3),> >> parity number,...)> >> that is provably nonHurwitz? And if Hurwitz, is not homographically> >> equivalent to a single-> >> mover, linear?> >>> >> Boy, are we ignorant.> >> --rwg
YOW! This paper has the general "single-mover" formula, and also has double movers! --rwg Out[584]= 1 Tanh[-------] Sqrt[2] ------------- Sqrt[2] In[585]:= hursigs2[%584, 99, 1] During evaluation of In[585]:= 0.006964 Out[585]= {1, 0 1 } 1 0 2 {3 + 4 k, 10 + 8 k} In[586]:= hursigs2[%584, 99, 2] During evaluation of In[586]:= 0.035164 Out[586]= 0 2 {3, 1 1 8 {1, 1, 8 k, 1, 1, 1 + 4 k, 20 + 32 k, 3 + 4 k}} 0 2 1 0 8 {1, 1, 3 + 4 k, 36 + 32 k, 5 + 4 k, 1, 1, 12 + 8 k} 0 1 2 0 2 {1 + 4 k, 6 + 8 k} In[587]:= hursigs2[%585, 99, 3] During evaluation of In[587]:= 0.082445 Out[587]= 0 3 1 2 14 {1, 2, 8 k, 2, 1, 4 k, 2, 1, 2 + 8 k, 1, 2, 2 + 4 k, 54 + 72 k, 3 + 4 k} {4, } 0 3 1 1 14 {1, 2, 3 + 4 k, 2, 1, 8 + 8 k, 45 + 36 k, 11 + 8 k, 2, 1, 5 + 4 k, 1, 2, 13 + 8 k} 0 3 1 0 10 {1, 3 + 4 k, 78 + 72 k, 5 + 4 k, 102 + 72 k, 6 + 4 k, 2, 1, 13 + 8 k, 2} 0 1 3 0 10 {1, 2, 4 k, 1, 2, 3 + 8 k, 21 + 36 k, 6 + 8 k, 33 + 36 k, 8 + 8 k} In[588]:= hursigs2[%585, 99, 4] During evaluation of In[588]:= 0.087023 Out[588]= 0 4 {7, 1 3 8 {1, 3, 4 k, 3, 1, 2 k, 40 + 64 k, 1 + 2 k} } 0 4 1 2 12 {1, 1, 2 k, 3, 1, 1 + 4 k, 1, 3, 1 + 2 k, 1, 1, 17 + 16 k} 0 4 1 1 12 {1, 1, 1 + 2 k, 3, 1, 3 + 4 k, 1, 3, 2 + 2 k, 1, 1, 25 + 16 k} 0 4 1 0 8 {1, 3, 2 + 4 k, 3, 1, 1 + 2 k, 72 + 64 k, 2 + 2 k} 0 2 2 1 8 {1, 1, 4 k, 12 + 32 k, 2 + 4 k, 1, 1, 6 + 8 k} 0 2 2 0 2 {3 + 4 k, 10 + 8 k} 0 1 4 0 8 {1, 1, 2 + 4 k, 28 + 32 k, 4 + 4 k, 1, 1, 10 + 8 k}
Oh foo, I've been had. This is a different Komatsu paper, http://www.researchgate.net/publication/225486369_Hurwitz_continued_fraction... and the double mover effect is a trivial consequence of the regular CF scaling rule, where you alternately multiply and divide. --rwg On Sun, Sep 30, 2012 at 11:22 PM, Bill Gosper <billgosper@gmail.com> wrote:
BG> By uncontrived I meant not explicitly constructed as a continued fraction.
NeilB>Just my two cents:http://ttk.pte.hu/mii/html/pannonica/index_elemei/mp17-1/mp17-1-091-110.pdf claims that (1;2,4,8,16...) is equal to
1+Sum[1/((-1)^n*2^(1 + n)^2*QPochhammer[4, 4, n]), {n, 0, Infinity}]/ Sum[1/((-1)^n*2^n^2*QPochhammer[4, 4, n]), {n, 0, Infinity}]
(The source, Takao Komatsu's "Hurwitz and Tasoev Continued Fractions with Long Period" cites another paper, "On Hurwitzian and Tasoev's Continued Fractions", by the same author, but which I can't seem to find) --Neil Bickford
On Thu, Sep 27, 2012 at 10:33 AM, Allan Wechsler <acwacw@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=acwacw%40gmail.com>> wrote:>> If x is Hurwitz then so is x/2, so, uh ...>> I think that by "contrived" Gosper means "with cf terms explicitly chosen> to be non-periodic". An example would be (1;2,4,8,16...). I'm assuming> that there is not a single such example that anybody knows a "closed form"> for; the closed form thereof would answer Gosper's query.>> On Thu, Sep 27, 2012 at 12:54 PM, meekerdb <meekerdb@verizon.net <http://gosper.org/webmail/src/compose.php?send_to=meekerdb%40verizon.net>> wrote:>> > Is the smallest contrived Hurwitz, thereby uncontrived? :-)> >> > Brent
> On 9/27/2012 5:38 AM, Bill Gosper wrote:> >> >> Can someone name a single (uncontrived) constant (e.g., π, e^3, 2^(1/3),> >> parity number,...)> >> that is provably nonHurwitz? And if Hurwitz, is not homographically> >> equivalent to a single-> >> mover, linear?> >>> >> Boy, are we ignorant. --rwg
YOW! This paper has the general "single-mover" formula, and also has double movers! --rwg
Out[584]= 1 Tanh[-------] Sqrt[2] ------------- Sqrt[2]
In[585]:= hursigs2[%584, 99, 1]
During evaluation of In[585]:= 0.006964
Out[585]= {1, 0 1 }
1 0 2 {3 + 4 k, 10 + 8 k}
In[586]:= hursigs2[%584, 99, 2]
During evaluation of In[586]:= 0.035164
Out[586]= 0 2
{3, 1 1 8 {1, 1, 8 k, 1, 1, 1 + 4 k, 20 + 32 k, 3 + 4 k}}
0 2
1 0 8 {1, 1, 3 + 4 k, 36 + 32 k, 5 + 4 k, 1, 1, 12 + 8 k}
0 1
2 0 2 {1 + 4 k, 6 + 8 k}
In[587]:= hursigs2[%585, 99, 3]
During evaluation of In[587]:= 0.082445
Out[587]= 0 3
1 2 14 {1, 2, 8 k, 2, 1, 4 k, 2, 1, 2 + 8 k, 1, 2, 2 + 4 k, 54 + 72 k, 3 + 4 k} {4, }
0 3
1 1 14 {1, 2, 3 + 4 k, 2, 1, 8 + 8 k, 45 + 36 k, 11 + 8 k, 2, 1, 5 + 4 k, 1, 2, 13 + 8 k}
0 3
1 0 10 {1, 3 + 4 k, 78 + 72 k, 5 + 4 k, 102 + 72 k, 6 + 4 k, 2, 1, 13 + 8 k, 2}
0 1
3 0 10 {1, 2, 4 k, 1, 2, 3 + 8 k, 21 + 36 k, 6 + 8 k, 33 + 36 k, 8 + 8 k}
In[588]:= hursigs2[%585, 99, 4]
During evaluation of In[588]:= 0.087023
Out[588]= 0 4
{7, 1 3 8 {1, 3, 4 k, 3, 1, 2 k, 40 + 64 k, 1 + 2 k} }
0 4
1 2 12 {1, 1, 2 k, 3, 1, 1 + 4 k, 1, 3, 1 + 2 k, 1, 1, 17 + 16 k}
0 4
1 1 12 {1, 1, 1 + 2 k, 3, 1, 3 + 4 k, 1, 3, 2 + 2 k, 1, 1, 25 + 16 k}
0 4
1 0 8 {1, 3, 2 + 4 k, 3, 1, 1 + 2 k, 72 + 64 k, 2 + 2 k}
0 2
2 1 8 {1, 1, 4 k, 12 + 32 k, 2 + 4 k, 1, 1, 6 + 8 k}
0 2
2 0 2 {3 + 4 k, 10 + 8 k}
0 1
4 0 8 {1, 1, 2 + 4 k, 28 + 32 k, 4 + 4 k, 1, 1, 10 + 8 k}
Bill Gosper <billgosper@gmail.com> wrote:
BG> By uncontrived I meant not explicitly constructed as a continued fraction.
NeilB>Just my two cents:http://ttk.pte.hu/mii/html/pannonica/index_elemei/mp17-1/mp17-1-091-110.pdf claims that (1;2,4,8,16...) is equal to
1+Sum[1/((-1)^n*2^(1 + n)^2*QPochhammer[4, 4, n]), {n, 0, Infinity}]/ Sum[1/((-1)^n*2^n^2*QPochhammer[4, 4, n]), {n, 0, Infinity}]
(The source, Takao Komatsu's "Hurwitz and Tasoev Continued Fractions with Long Period" cites another paper, "On Hurwitzian and Tasoev's Continued Fractions", by the same author, but which I can't seem to find) --Neil Bickford
On Thu, Sep 27, 2012 at 10:33 AM, Allan Wechsler <acwacw@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=acwacw%40gmail.com>> wrote:>> If x is Hurwitz then so is x/2, so, uh ...>> I think that by "contrived" Gosper means "with cf terms explicitly chosen> to be non-periodic". An example would be (1;2,4,8,16...). I'm assuming> that there is not a single such example that anybody knows a "closed form"> for; the closed form thereof would answer Gosper's query.>> On Thu, Sep 27, 2012 at 12:54 PM, meekerdb <meekerdb@verizon.net <http://gosper.org/webmail/src/compose.php?send_to=meekerdb%40verizon.net>> wrote:>> > Is the smallest contrived Hurwitz, thereby uncontrived? :-)> >> > Brent> >> >> > On 9/27/2012 5:38 AM, Bill Gosper wrote:> >>
Can someone name a single (uncontrived) constant (e.g., ?, e^3, 2^(1/3),> >> parity number,...)> >> that is provably nonHurwitz? And if Hurwitz, is not homographically> >> equivalent to a single-> >> mover, linear?> >>> >> Boy, are we ignorant.> >> --rwg
YOW! This paper has the general "single-mover" formula, and also has double movers! --rwg
Out[584]= 1 Tanh[-------] Sqrt[2] ------------- Sqrt[2]
In[585]:= hursigs2[%584, 99, 1]
During evaluation of In[585]:= 0.006964
Out[585]= {1, 0 1 }
1 0 2 {3 + 4 k, 10 + 8 k}
In[586]:= hursigs2[%584, 99, 2]
During evaluation of In[586]:= 0.035164
Out[586]= 0 2
{3, 1 1 8 {1, 1, 8 k, 1, 1, 1 + 4 k, 20 + 32 k, 3 + 4 k}}
0 2
1 0 8 {1, 1, 3 + 4 k, 36 + 32 k, 5 + 4 k, 1, 1, 12 + 8 k}
0 1
2 0 2 {1 + 4 k, 6 + 8 k}
In[587]:= hursigs2[%585, 99, 3]
During evaluation of In[587]:= 0.082445
Out[587]= 0 3
1 2 14 {1, 2, 8 k, 2, 1, 4 k, 2, 1, 2 + 8 k, 1, 2, 2 + 4 k, 54 + 72 k, 3 + 4 k} {4, }
0 3
1 1 14 {1, 2, 3 + 4 k, 2, 1, 8 + 8 k, 45 + 36 k, 11 + 8 k, 2, 1, 5 + 4 k, 1, 2, 13 + 8 k}
0 3
1 0 10 {1, 3 + 4 k, 78 + 72 k, 5 + 4 k, 102 + 72 k, 6 + 4 k, 2, 1, 13 + 8 k, 2}
0 1
3 0 10 {1, 2, 4 k, 1, 2, 3 + 8 k, 21 + 36 k, 6 + 8 k, 33 + 36 k, 8 + 8 k}
In[588]:= hursigs2[%585, 99, 4]
During evaluation of In[588]:= 0.087023
Out[588]= 0 4
{7, 1 3 8 {1, 3, 4 k, 3, 1, 2 k, 40 + 64 k, 1 + 2 k} }
0 4
1 2 12 {1, 1, 2 k, 3, 1, 1 + 4 k, 1, 3, 1 + 2 k, 1, 1, 17 + 16 k}
0 4
1 1 12 {1, 1, 1 + 2 k, 3, 1, 3 + 4 k, 1, 3, 2 + 2 k, 1, 1, 25 + 16 k}
0 4
1 0 8 {1, 3, 2 + 4 k, 3, 1, 1 + 2 k, 72 + 64 k, 2 + 2 k}
0 2
2 1 8 {1, 1, 4 k, 12 + 32 k, 2 + 4 k, 1, 1, 6 + 8 k}
0 2
2 0 2 {3 + 4 k, 10 + 8 k}
0 1
4 0 8 {1, 1, 2 + 4 k, 28 + 32 k, 4 + 4 k, 1, 1, 10 + 8 k} I wonder if the old papers of D.H. Lehmer might be of interest in some problems here? His complete bibliography is in Acta Arithmetica LXII.3 (1992). Here are two of his papers:
Arithmetic Periodicites of Bessel Functions, Annals of Math., Vol. 43(1932), 143-150. Continued Fractions Containing Arithmetic Progressions, Scripta Mathematica, V. XXIX. Nos. 1,2. John
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